Nm

X ai X xNjxij =XxNjyj i j j

These N equations with N unknowns can also be written in a matrix-vector notation.

The formal solution can be obtained by multiplying with the inverse X matrix.

In actual applications, the X matrix may be singular, or nearly so, and the inverse matrix either does not exist or is prone to numerical errors. A singular matrix indicates that at least one of the linear equations can be written as a combination of the other equations, and such cases can be handled by singular value decomposition methods, as discussed in Section 16.2. Indeed, for the example of determining partial charges by fitting to the electrostatic potential, the equations determining the charges on the atoms far from the molecular surface are often poorly conditioned, i.e. the external electrostatic potential is only weakly dependent on the charges on the buried atoms.

Other examples of optimizing functions that depend quadratically of the parameters include the energy of Hartree-Fock (HF) and configuration interaction (CI) wave functions. Minimization of the energy with respect to the MO or CI coefficients leads to a set of linear equations. In the HF case, the xij coefficients depend on the parameters ai, and must therefore be solved iteratively. In the CI case, the number of parameters is typically 106-109, and a direct solution of the linear equations is therefore prohibitive, and special iterative methods are used instead. The use of iterative techniques for solving the CI equations is not due to the mathematical nature of the problem, but due to computational efficiency considerations.

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